Diffusiophoresis and Diffusio-osmosis into a Dead-End Channel: Role of the Concentration-Dependence of Zeta Potential

Chemically induced transport methods open up new opportunities for colloidal transport in dead-end channel geometries. Diffusiophoresis, which describes particle movement under an electrolyte concentration gradient, has previously been demonstrated in dead-end channels. The presence of solute concentration gradients in such channels induces particle motion (phoresis) and fluid flow along solid walls (osmosis). The particle velocity inside a dead-end channel is thus influenced by particle diffusiophoresis and wall diffusio-osmosis. The magnitude of phoresis and osmosis depends on the solute’s relative concentration gradient, the electrokinetic parameters of the particle and the wall, and the diffusivity contrast of cations and anions. Although it is known that some of those parameters are affected by electrolyte concentration, e.g., zeta potential, research to date often interprets results using averaged and constant zeta potential values. In this work, we demonstrate that concentration-dependent zeta potentials are essential when the zeta potential strongly depends on electrolyte concentration for correctly describing the particle transport inside dead-end channels. Simulations including concentration-dependent zeta potentials for the particle and wall matched with experimental observations, whereas simulations using constant, averaged zeta potentials failed to capture particle dynamics. These results contribute to the fundamental understanding of diffusiophoresis and the diffusio-osmosis process.


Supporting Information Available 1 Particle Properties
Two particles were used in this study. The general information for each particle is given below.

Zeta Potential of Particles
The zeta potential was measured in the 1-10 mM NaCl range and given in Figure S1. The equation (ζ(c N aCl ) = a + b log 10 (c N aCl )) was fitted for each particle's zeta potential value.
The constants a and b were calculated as -81.61 mV and 15.61 mV for PS-carboxylate, and -27.43 mV and 22.63 mV for PS-PEG particles (c N aCl is in mM).
The zeta potential change depends on the solute relative gradient, and the b = b/2.3026 values (EquationS2).
Since the solute relative gradient is the same, b value determines the value of the zeta potential change. Figure S2 suggested that the zeta potential change is high at the dead-end channel entrance and reducing in time.  Figure S1: Zeta Potential of particles in NaCl solutions. Two different particles, PScarboxylate, and PS-PEG were used in this study. The equation (ζ = a + b log 10 c N aCl )) was fitted for each particle's zeta potential value. The error bar shows the standard deviation of three samples.

Effect of Gravity
The particles are significant in size (≈ 1 µm) compared to the channel height (10 µm).
Even though the density difference between the particle and water (∆ρ = 50 kg/m 3 ) is low, particles tend to move towards the bottom in time. When the particles start in the middle of the channel, they move to the bottom wall in about 2-3 minutes. However, the electrokinetic force and wall exclusion effect prevent particles from sticking to the bottom. Moreover, the confocal microscope images of similar size particles 1 show the effect of gravity is not extreme.

Zeta Potential of Polydimethylsiloxane (PDMS)
The zeta potential of PDMS was calculated by streaming potential measurements in 0.1 -10 mM NaCl. The zeta potential results were compared with literature (ζ = a + b log 10 (c) with a=6.27 mV and b=29.75 mV). 2 A similar trend is observed ( Figure S3). Values for a=6.27 mV and b=29.75 mV were employed in the simulations. Figure S4 showed that the zeta potential change is high at the dead-end channel entrance and reducing in time.

Experimental Error Sources
There are three primary sources of error due to particle detection and calculation methods.
Firstly, the particle position might not be detected precisely due to the finite size of pixels in the camera, and this uncertainty is estimated by. 3 where ∆V is velocity uncertainity, and ∆t is the image time frame. ∆x is the uncertainity position and estimated as micron to pixel ratio over how many pixels are used for detect a particle (= 0.327/7 µm). This source of error is inversely proportional to the image frame  rate and calculated for 10 fps as 0.66 µm/s, whereas for 1 fps as 0.06 µm/s. Therefore, this error source is relatively small.
Secondly, Brownian motion and the particle velocity might be of the same magnitude when the particle velocity is small (< 1 µm/s). The relative error is determined by the following formula that describes the relative error in the PIV system: 4 where U is the particle's velocity, and D p is the diffusivity of the particle. We determined Figure S4: PDMS Zeta Potential Change according to x-Location.
the limit which is the error, and the velocity magnitudes are equal ( x = U ). These limits are 1.7 µm/s for 10 fps, and 0.96 µm/s for 1 fps. Velocity magnitudes below these limits might be due to Brownian motion.
Thirdly, the response of particles under sudden acceleration becomes important in low viscosity. The response time is inversely proportioned to environmental viscosity, and it is estimated by; 4 The response time is 78 ns which is much lower than the image frame time. It is negligible in the analysis.

Particle Velocityy component
The highest movement in the particle is observed in the x-direction. y component of the absolute velocity value shows particles do not perform a lateral movement ( Figure S7). We only observed a relatively high-velocity profile at the entrance of the dead-end channel. After ≈ 20 > µm, we only observed an error in the particle velocity due to Brownian motion. Our error calculation for Brownian motion for the PIV case (for 5fps ≈ 1.15 µm/s) also supports this observation. Figure S7: Absolute y-component of particle velocity for PS-carboxylate particle at 60 seconds. In the calculation of the particle velocities, we reduced the frame rate to 5fps. The enclosed area shows the standard error calculated according to the number of particles.
Approximately 300-400 particles were tracked in one experimental study. Figure 4 shows the average of those particles in certain locations in the 1-D domain. Each particle's x component velocity value is given according to the x direction. The range given in Figure S8 (light and dark blue lines) were calculated by combining the particle diffusiophoresis, and fluid flow and we performed calculations for concentration-independent ( Figure S8A) and concentration-dependent zeta potential values( Figure S8B). Almost all the particle trajectories were captured when the concentration-dependent zeta potential values were used.

Surface-induced Flow
Surface-induced flow on the channel wall creates convective flow inside the dead-end channel.
However, the magnitude of the center line convective velocity is not equal to the magnitude of the slip velocity (u slip = −u centerline ). Figure S9 shows the x-component velocity profile at the y-z plane. As can be seen from the figure, the velocity magnitude near the wall is different from the middle suggesting that the slip velocity is higher compared to the centerline velocity.
We determined centerline velocity is ≈ 0.72 of the slip velocity (u slip = −0.72 · u centerline ). Figure S9: y-z cross-plane velocity profile at x = 50µm and t= 60 s.The color bar is given below the image, and the unit is µm/s.

PEG Zeta Potential and Wall Mobility Change
We performed the particle simulation for the constant zeta potential values for the wall (ζ P DM S = −60mV ) and concentration-dependent zeta potential value for the PS-PEG particles. We also quantify behavior in Figure S10 which shows the zeta potential and mobility differences compared to their constant values. Figure S10: Effect of using concentration-dependent zeta potential. (A) Zeta potential difference between the constant ζ P S−P EG = -10 mV & ζ P DM S = -60 mV, and the zeta potential value where the front particle position. (B) Mobility difference between the average Γ P S−P EG = 4.59 × 10 −11 m 2 /s & Γ P DM S = 5.24 × 10 −10 m 2 /s, and the mobility value where the front particle position.

Theoretical Velocity Profiles
We showed the theoretical velocity profile of convective flow and diffusiophoresis at 60 seconds below. We calculated the centerline velocity according to front particle position by considering constant and concentration-dependent zeta potential cases. 14 Video S1 and S2 Video S1 is the video of Figure 6 A,B and C. Video S2 is the video of Figure 8 A,B and C.
For each video, play speed is 10x (0-300 seconds) and scale bar is 50 µm.